Calculus: Continuity, Discontinuity, and Derivatives

Vocabulary flashcards covering limits, continuity conditions, discontinuity examples, secant line slope, and tangent lines from Chapters 4 and 5.

Left Hand Limit (L.H.L)

The value that a function $f(x)$ approaches as $x$ approaches a value $a$ from the left side, represented mathematically as $\lim_{x \to a^-} f(x)$.

Right Hand Limit (R.H.L)

The value that a function $f(x)$ approaches as $x$ approaches a value $a$ from the right side, represented mathematically as $\lim_{x \to a^+} f(x)$.

Limit Existence

The condition where a function's left hand limit equals its right hand limit at a given point, expressed as $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.

Continuity Condition

The requirement that a limit exists at $x = a$ and equals the defined value of the function at that point, expressed as $\lim_{x \to a} f(x) = f(a)$.

Salary-Sales Discontinuity Example

A pay function with a basic amount of $800$, $10\%$ commission, and a lump-sum bonus of $500$ for sales $S \ge 20,000$; it is discontinuous at $S = 20,000$ because the left hand limit ($2800$) does not equal the right hand limit ($3300$).

Secant Line

A line passing through two points on the curve of a function $f(x)$.

Secant Slope

The slope of a secant line connecting two points on a curve, calculated using the formula $\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$.

Delta y ($\Delta y$)

The change in the dependent variable between two points on a function, defined as $\Delta y = f(x_2) - f(x_1)$.

Tangent Line

A straight line that touches the graph of a function $f(x)$ at a single point.